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Positivity

, Volume 23, Issue 4, pp 853–874 | Cite as

Characterization of Strict Positive Definiteness on products of complex spheres

  • Mario H. Castro
  • Eugenio Massa
  • Ana Paula PeronEmail author
Article
  • 77 Downloads

Abstract

In this paper we consider Positive Definite functions on products \(\Omega _{2q}\times \Omega _{2p}\) of complex spheres, and we obtain a condition, in terms of the coefficients in their disc polynomial expansions, which is necessary and sufficient for the function to be Strictly Positive Definite. The result includes also the more delicate cases in which p and/or q can be 1 or \(\infty \). The condition we obtain states that a suitable set in \({\mathbb {Z}}^2\), containing the indexes of the strictly positive coefficients in the expansion, must intersect every product of arithmetic progressions.

Keywords

Strictly Positive Definite functions Product of complex spheres Generalized Zernike polynomial 

Mathematics Subject Classification

42A82 42C10 

Notes

Acknowledgements

Mario H. Castro was supported by: Grant \(\#\)APQ-00474-14, FAPEMIG and CNPq/Brazil. Eugenio Massa was supported by: Grant \(\#\)2014/25398-0, São Paulo Research Foundation (FAPESP) and Grant \(\#\)303447/2017-6, CNPq/Brazil. Ana P. Peron was supported by: Grants \(\#\)2016/03015-7, \(\#\)2016/09906-0 and \(\#\)2014/25796-5, São Paulo Research Foundation (FAPESP).

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Departamento de MatemáticaUFUUberlândiaBrazil
  2. 2.Departamento de MatemáticaICMC-USP - São CarlosSão CarlosBrazil

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